It was shown in [1]} that \(\mathrm {Cu}(A)\)  admits suprema for increasing sequences. We say that \([a]\)  is compactly contained in \([b]\) , written \([a] \ll [b]\) , if whenever \(([b_i])\)  is increasing and \(\sup [b_i] = [b]\)  we have \([a] \le [b_{i_0}]\)  for some \(i_0 \in \mathbb {N}\) . An element \([a]\)  of \(\mathrm {Cu}(A)\)  is called compact if \([a] \ll [a]\) . If \(A\)  is simple or of stable rank one, then \([a]\)  is compact if and only if \(a \sim p\)  for some projection \(p\)  ([2]}, [3]}). Recall that \(A\)  has real rank zero if every self-adjoint element in \(A\)  can be approximated in norm by self-adjoint elements with finite spectrum. Consequently, for such algebras, each \([a] \in \mathrm {Cu}(A)\)  is seen to be the supremum of a sequence of compact elements by exploiting (i) below.
